Mathematicians have conjectured since Euclid’s time that there are infinite pairs of prime numbers separated from each other by 2. Despite the fact that primes are separated on average by bigger gaps as numbers increase, evidence suggests that primes continue to appear as “twin primes” (green triangles) no matter how high you go. The illustration above highlights prime numbers, counting from 1 at upper left to 300 at lower right. Below, prime numbers are shown, counting from 1 at upper left to 625 at lower right.

But for mathematicians, suggestive evidence isn’t good enough. For at least a century, they’ve labored to prove the twin prime conjecture. A major advance came this spring, when University of New Hampshire mathematician Yitang Zhang showed that there are infinitely many primes separated by some number smaller than 70 million. That may be a lot bigger than their eventual goal of 2, but by the end of July mathematicians had already whittled that limit down to 5,414.

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